Efficient differentiable quadratic programming layers: an ADMM approach

Abstract

We present SCQPTH: a differentiable first-order splitting method for convex quadratic programs. The SCQPTH framework is based on the alternating direction method of multipliers (ADMM) and the software implementation is motivated by the state-of-the art solver OSQP: an operating splitting solver for convex quadratic programs (QPs). The SCQPTH software is made available as an open-source python package and contains many similar features including efficient reuse of matrix factorizations, infeasibility detection, automatic scaling and parameter selection. The forward pass algorithm performs operator splitting in the dimension of the original problem space and is therefore suitable for large scale QPs with 100 − 1000 decision variables and thousands of constraints. Backpropagation is performed by implicit differentiation of the ADMM fixed-point mapping. Experiments demonstrate that for large scale QPs, SCQPTH can provide a 1 × −10× improvement in computational efficiency in comparison to existing differentiable QP solvers.

Novel Multivariable Evolutionary Algorithm-Based Method for Modal Reconstruction of the Corneal Surface from Sparse and Incomplete Point Clouds

“…In clinical practice, modal methods used are based on Zernike polynomials, which have shown high accuracy in slightly deformed corneas and good robustness against noise present in equipment during the measurement acquisition process [ 3 ], which gives them a lower dependence on measurement acquisition errors [ 14 ]. However, these polynomials have some problems due to their global nature, that is, in corneas that present significant surface irregularities, such as in the case of advanced keratoconus; these polynomials require high orders to perform a reliable reconstruction of corneal geometry, for which they use fitting tools such as least squares (LSQ) [ 15 ], or sequential quadratic programming (SQP) [ 16 ], but both generate instabilities against local minima caused by the discontinuities as mentioned above [ 17 , 18 , 19 ]. Therefore, it would be of interest to develop a modal reconstruction procedure that is not only accurate when irregular surfaces are present but also computationally viable in clinical practice.…”